Problem: Solve for $x$ : $ 5|x - 2| - 9 = 1|x - 2| + 8 $
Solution: Subtract $ {1|x - 2|} $ from both sides: $ \begin{eqnarray} 5|x - 2| - 9 &=& 1|x - 2| + 8 \\ \\ { - 1|x - 2|} && { - 1|x - 2|} \\ \\ 4|x - 2| - 9 &=& 8 \end{eqnarray} $ Add ${9}$ to both sides: $ \begin{eqnarray} 4|x - 2| - 9 &=& 8 \\ \\ { + 9} &=& { + 9} \\ \\ 4|x - 2| &=& 17 \end{eqnarray} $ Divide both sides by ${4}$ $ \dfrac{4|x - 2|} {{4}} = \dfrac{17} {{4}} $ Simplify: $ |x - 2| = \dfrac{17}{4}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 2 = -\dfrac{17}{4} $ or $ x - 2 = \dfrac{17}{4} $ Solve for the solution where $x - 2$ is negative: $ x - 2 = -\dfrac{17}{4} $ Add ${2}$ to both sides: $ \begin{eqnarray} x - 2 &=& -\dfrac{17}{4} \\ \\ {+ 2} && {+ 2} \\ \\ x &=& -\dfrac{17}{4} + 2 \end{eqnarray} $ Change the ${ + 2}$ to an equivalent fraction with a denominator of $4$ $ x = - \dfrac{17}{4} {+ \dfrac{8}{4}} $ $ x = -\dfrac{9}{4} $ Then calculate the solution where $x - 2$ is positive: $ x - 2 = \dfrac{17}{4} $ Add ${2}$ to both sides: $ \begin{eqnarray} x - 2 &=& \dfrac{17}{4} \\ \\ {+ 2} && {+ 2} \\ \\ x &=& \dfrac{17}{4} + 2 \end{eqnarray} $ Change the ${ + 2}$ to an equivalent fraction with a denominator of $4$ $ x = \dfrac{17}{4} {+ \dfrac{8}{4}} $ $ x = \dfrac{25}{4} $ Thus, the correct answer is $x = -\dfrac{9}{4} $ or $x = \dfrac{25}{4} $.